3.1472 \(\int \frac {x^7}{1-x^8} \, dx\)

Optimal. Leaf size=12 \[ -\frac {1}{8} \log \left (1-x^8\right ) \]

[Out]

-1/8*ln(-x^8+1)

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {260} \[ -\frac {1}{8} \log \left (1-x^8\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^7/(1 - x^8),x]

[Out]

-Log[1 - x^8]/8

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin {align*} \int \frac {x^7}{1-x^8} \, dx &=-\frac {1}{8} \log \left (1-x^8\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ -\frac {1}{8} \log \left (1-x^8\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(1 - x^8),x]

[Out]

-1/8*Log[1 - x^8]

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fricas [A]  time = 0.51, size = 8, normalized size = 0.67 \[ -\frac {1}{8} \, \log \left (x^{8} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-x^8+1),x, algorithm="fricas")

[Out]

-1/8*log(x^8 - 1)

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giac [A]  time = 0.16, size = 9, normalized size = 0.75 \[ -\frac {1}{8} \, \log \left ({\left | x^{8} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-x^8+1),x, algorithm="giac")

[Out]

-1/8*log(abs(x^8 - 1))

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maple [B]  time = 0.01, size = 30, normalized size = 2.50 \[ -\frac {\ln \left (x -1\right )}{8}-\frac {\ln \left (x +1\right )}{8}-\frac {\ln \left (x^{2}+1\right )}{8}-\frac {\ln \left (x^{4}+1\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(-x^8+1),x)

[Out]

-1/8*ln(x-1)-1/8*ln(x+1)-1/8*ln(x^2+1)-1/8*ln(x^4+1)

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maxima [A]  time = 1.06, size = 8, normalized size = 0.67 \[ -\frac {1}{8} \, \log \left (x^{8} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-x^8+1),x, algorithm="maxima")

[Out]

-1/8*log(x^8 - 1)

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mupad [B]  time = 0.04, size = 8, normalized size = 0.67 \[ -\frac {\ln \left (x^8-1\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^7/(x^8 - 1),x)

[Out]

-log(x^8 - 1)/8

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sympy [A]  time = 0.29, size = 8, normalized size = 0.67 \[ - \frac {\log {\left (x^{8} - 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(-x**8+1),x)

[Out]

-log(x**8 - 1)/8

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